Sheaves of Abelian l-groups

Abstract

Keimel was first to associate a spectrum with an abelian l-group G. An alternative construction of a spectrum is proposed here. Its points are the prime l-ideals of G, i.e., they are the points of Keimel’s spectrum plus the improper prime l-ideal G. The spectrum is equipped with the inverse topology, which is different from the one used by Keimel. The new spectrum is denoted by Specinv(G) and is called the inverse spectrum of G. There is a natural construction of a sheaf of l-groups on Specinv(G), which is called the structure sheaf of G on Specinv(G). The inverse spectrum and its structure sheaf first appeared in Rump and Yang (Bull Lond Math Soc 40:263–273, 2008) via the Jaffard-Ohm Theorem. We present an elementary and direct construction. The relationship between an l-group H and an l-subgroup G is analyzed via the structure sheaves on the inverse spectra. The structure sheaves are used to give a sheaf-theoretic presentation of the valuation theory of a field. The group of Cartier divisors of an integral affine scheme is identified canonically with an l-group. Special attention is paid to Prufer domains, where the ties between rings and l-groups are particularly close.